Girlfriend never offers to pay

Hello everyone. Today I'm asking advice for my father (73 yrs), who's been in a new relationship for about 8-12 months with someone of similar age.

I'll tell you what I know. They live about 30-40 minute drive apart, and when they get together they usually play cards for hours on end or watch a movie(s) in his living room. Sometimes though they go out for dinner or go for a Miata run, from a club my father is in.

He's always the one making and suggesting the plans, and she never seems to offer to pitch in. Yesterday they went on a Miata run and never offered any money on gas, and when out for a meal just says 'I'll have what your having' or just gets a simple small salad. She never orders anything expensive on the menu.
He was considering doing a $500 Miata run that they are doing, but knows there are hotels and meals and gas to pay for and was asking for my help in what he should do and if my friends offer to pay for gas, which I replied that they all offer up money, male and female.
She doesn't seem to have any debit or broke, she seems to be self sufficient. So I thought she might have that old fashioned mind frame where the man makes the plan and pays for everything.

I suggested he could bring up the idea of the run, that would be fun and politely suggest he'd need her to pitch in 35-50%. Any advice?

Thank you